I'm hoping for feedback on my understanding of how to approach this question.
Firstly to recall;
$$R = \mathbb Z[x]/\langle x^2 + 1\rangle = \{ l(x) + \langle x^2 + 1\rangle | l(x) = 0 \text{ or } deg(l) < 2\}$$
If $a(x),b(x) \in R, $ we define multiplication and addition as:
$$a(x)\cdot b(x) = a(x) \cdot b(x) \text{ mod } x^2 +1,\ a(x) + b(x) = a(x) + b(x) \text{ mod } x^2 +1$$
To understand how this structure works we can see what happens to the basis on multiplication:
$$1\cdot1 = 0\cdot(x^2 + 1) + 1 = 1,\ x\cdot 1 = 1\cdot x = 0\cdot(x^2 + 1) + 1 = 1,\ x\cdot x = 1\cdot(x^2 + 1) - 1 = -1$$
So by investigation I can treat the $1$ as the real part and $x$ as the imaginary part. Given an arbitrary element of $R$ has the form $ \alpha \cdot 1 + \beta \cdot x$ for $\alpha,\beta\in \mathbb Z$, I define the mapping $\phi:R\rightarrow R$ as $\phi(\alpha \cdot 1 + \beta \cdot x) \rightarrow \alpha + \beta I$. Now to check this is an isomorphism between the two rings, I need to check that the mapping is a bijection on both operations of the ring.
I won't do this part as it's quite long winded, but my general approach here highlights a correct understanding right?
Thanks heaps.
